Does anybody know an easy explanation of the proof of Artin's vanishing theorem (that the etale cohomology of an affine variety of dimension $n$ over an algebraically closed field vanishes in degrees $>n$, or of any other version of this statement)? I have found some proofs; all of them are step by step, and it is not clear to me which of these steps are the most important ones. So, what is the central idea here?
I was curious myself after learning this result sometime ago from Lazarsfeld's book on positivity (he calls it the Artin-Grothendieck theorem). The corresponding statement for smooth varieties over the complex numbers and singular cohomology (theorem of Andreotti-Frankel) follows from the fact that Morse theory shows that the variety is homotopy equivalent to a CW-complex with no cells in dimensions $>n$.
The etale cohomology counterpart works more generally for constructible sheaves. This is probably not very helpful, but here is a sketch of the argument from Lazarsfeld (he uses constructible sheaves in the complex topology, but it should adapt to etale sheaves):
So all in all it is a typical example of devissage, which I usually to dislike but slowly learn to appreciate. I think from the outline it is clear which are the key ideas, but I would still really like to see a conceptual proof.